Abstract
We prove a Lie-theoretic result for type A root systems. This allows us to determine the unipotent orbits attached to degenerate Eisenstein series on general linear groups, confirming a conjecture of David Ginzburg. In particular, this also shows that any unipotent orbit of general linear groups does occur as the unipotent orbit attached to a specific automorphic representation. The proof of the Lie-theoretic result relies on the notion of the descending decomposition, which expresses every Weyl group element as a product of simple reflections in a certain way. It is suitable for induction and allows us to translate the question into a combinatorial statement.
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References
Bernstein, I.N., Zelevinsky, A.V.: Induced representations of reductive \({\mathfrak{p}}\)-adic groups. I. Ann. Sci. École Norm. Sup. (4) 10, 441–472 (1977)
Bump, D.: Lie Groups, vol. 225 of Graduate Texts in Mathematics, 2nd edn. Springer, New York (2013)
Cai, Y.: Fourier coefficients for Theta representations on covers of general linear groups. Trans. Amer. Math. Soc. (2016). arXiv:1602.06614. doi:10.1090/tran/7429 (to appear)
Carter, R.W.: Finite groups of Lie type, Pure and Applied Mathematics (New York). Wiley, New York (1985). Conjugacy classes and complex characters, A Wiley-Interscience Publication
Casselman, W.: Introduction to the theory of admissible representations of \(p\)-adic reductive groups, unpublished notes
Casselman, W., Shalika, J.: The unramified principal series of \(p\)-adic groups. II. The Whittaker function. Compos. Math. 41, 207–231 (1980)
Collingwood, D.H., McGovern, W.M.: Nilpotent Orbits in Semisimple Lie Algebras, Van Nostrand Reinhold Mathematics Series. Van Nostrand Reinhold Co., New York (1993)
Fleig, P., Gustafsson, H.P., Kleinschmidt, A., Persson, D.: Eisenstein series and automorphic representations (2015). arXiv:1511.04265
Ginzburg, D.: Certain conjectures relating unipotent orbits to automorphic representations. Isr. J. Math. 151, 323–355 (2006)
Ginzburg, D.: Eulerian integrals for \({\rm GL}_n\). In: Multiple Dirichlet Series, Automorphic Forms, and Analytic Number Theory, vol. 75 of Proceedings of Symposium Pure Mathematics, American Mathematical Society, Providence, RI, pp. 203–223 (2006)
Ginzburg, D.: Towards a classification of global integral constructions and functorial liftings using the small representations method. Adv. Math. 254, 157–186 (2014)
Ginzburg, D., Rallis, S., Soudry, D.: On Fourier coefficients of automorphic forms of symplectic groups. Manuscr. Math. 111, 1–16 (2003)
Ginzburg, D., Rallis, S., Soudry, D.: The Descent Map from Automorphic Representations of \({\rm GL}(n)\) to Classical Groups. World Scientific Publishing Co Pte. Ltd., Hackensack, NJ (2011)
Gomez, R., Gourevitch, D., Sahi, S.: Generalized and degenerate Whittaker models. Compos. Math. 153, 223–256 (2017)
Green, M.B., Miller, S.D., Russo, J.G., Vanhove, P.: Eisenstein series for higher-rank groups and string theory amplitudes. Commun. Number Theory Phys. 4, 551–596 (2010)
Gustafsson, H.P.A., Kleinschmidt, A., Persson, D.: Small automorphic representations and degenerate Whittaker vectors. J. Number Theory 166, 344–399 (2016)
Jiang, D.: Automorphic integral transforms for classical groups I: endoscopy correspondences. In: Automorphic Forms and Related Geometry: Assessing the Legacy of I. I. Piatetski-Shapiro, vol. 614 of Contemporary Mathematics, American Mathematical Society, Providence, RI, pp. 179–242 (2014)
Jiang, D., Liu, B.: On Fourier coefficients of automorphic forms of \({\rm GL}(n)\). Int. Math. Res. Not. IMRN, 4029–4071 (2013)
Jiang, D., Liu, B.: Arthur parameters and Fourier coefficients for automorphic forms on symplectic groups. Ann. Inst. Fourier 66, 477–519 (2016)
Jiang, D., Liu, B., Savin, G.: Raising nilpotent orbits in wave-front sets. Represent Theory 20, 419–450 (2016)
Littelmann, P.: Cones, crystals, and patterns. Transform. Groups 3, 145–179 (1998)
Matumoto, H.: Whittaker vectors and associated varieties. Invent. Math. 89, 219–224 (1987)
Miller, S.D., Sahi, S.: Fourier coefficients of automorphic forms, character variety orbits, and small representations. J. Number Theory 132, 3070–3108 (2012)
Mœglin, C.: Front d’onde des représentations des groupes classiques \(p\)-adiques. Am. J. Math. 118, 1313–1346 (1996)
Mœglin, C., Waldspurger, J.-L.: Modèles de Whittaker dégénérés pour des groupes \(p\)-adiques. Math. Z. 196, 427–452 (1987)
Reeder, M.: Notes on group theory. https://www2.bc.edu/mark-reeder/Groups.pdf
Shahidi, F.: Whittaker models for real groups. Duke Math. J. 47, 99–125 (1980)
Zelevinsky, A.V.: Induced representations of reductive \({\mathfrak{p}}\)-adic groups. II. On irreducible representations of \({\rm GL}(n)\). Ann. Sci. École Norm. Sup. (4) 13, 165–210 (1980)
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Cai, Y. Fourier coefficients for degenerate Eisenstein series and the descending decomposition. manuscripta math. 156, 469–501 (2018). https://doi.org/10.1007/s00229-017-0984-x
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DOI: https://doi.org/10.1007/s00229-017-0984-x